Based on the Lagrangian extremum law with the constraint that rotation matrix is an orthonormal matrix, the paper presents a new analytical algorithm of weighted 3D datum transformation. It is a stepwise algorithm. Firstly, the rotation matrix is computed using eigenvalue-eigenvector decomposition. Then, the scale parameter is computed with computed rotation matrix. Lastly, the translation parameters are computed with computed rotation matrix and scale parameter. The paper investigates the stability of the presented algorithm in the cases that the common points are distributed in 3D, 2D, and 1D spaces including the approximate 2D and 1D spaces, and gives the corresponding modified formula of rotation matrix. The comparison of the presented algorithm and classic Procrustes algorithm is investigated, and an improved Procrustes algorithm is presented since that the classic Procrustes algorithm may yield a reflection rather than a rotation in the cases that the common points are distributed in 2D space. A simulative numerical case and a practical case are illustrated.

Three-dimensional coordinate transformation problem is the most frequent problem in photogrammetry, geodesy, mapping, geographical information science (GIS), and computer vision. To overcome the drawback that traditional solution of the problem based on rotation angles depends strongly on initial value of parameter, which makes the method ineffective in the case of super-large rotation angle, the paper adopts an unit quaternion to represent three-dimensional rotation matrix, then puts forward a quaternion-based iterative solution of the problem. The cases study shows that the quaternion-based solution has no dependence on the initial value of parameter and desirable result with fast speed. Thus it is valid for three-dimensional coordinate transformation of any rotation angle

In this study, an approach using ground control point-free unmanned aerial vehicle (UAV)-based photogrammetry is proposed to estimate the volume of stockpiles carried on barges in a dynamic environment. Compared with similar studies regarding UAVs, an indirect absolute orientation based on the geometry of the vessel is used to establish a custom-built framework that can provide a unified reference instead of prerequisite ground control points (GCPs). To ensure sufficient overlap and reduce manual intervention, the stereo images are extracted from a UAV video for aerial triangulation. The region of interest is defined to exclude the area of water in all UAV images using a simple linear iterative clustering algorithm, which segments the UAV images into superpixels and helps to improve the accuracy of image matching. Structure-from-motion is used to recover three-dimensional geometry from the overlapping images without assistance of exterior parameters obtained from the airborne global positioning system and inertial measurement unit. Then, the semi-global matching algorithm is used to generate stockpile-covered and stockpile-free surface models. These models are oriented into a custom-built framework established by the known distance, such as the length and width of the vessel, and they do not require GCPs for coordinate transformation. Lastly, the volume of a stockpile is estimated by multiplying the height difference between the stockpile-covered and stockpile-free surface models by the size of the grid that is defined using the resolution of these models. Results show that a relatively small deviation of approximately ±2% between the volume estimated by UAV photogrammetry and the volume calculated by traditional manual measurement was obtained. Therefore, the proposed approach can be considered the better solution for the volume measurement of stockpiles carried on barges in a dynamic environment because UAV-based photogrammetry not only attains superior density and spatial object accuracy but also remarkably reduces data collection time.

Rigid transformation including rotation and translation can be elegantly represented by a unit dual quaternion. Thus, a non-differential model of the Helmert transformation (3D seven-parameter similarity transformation) is established based on unit dual quaternion. This paper presents a rigid iterative algorithm of the Helmert transformation using dual quaternion. One small rotation angle Helmert transformation (actual case) and one big rotation angle Helmert transformation (simulative case) are studied. The investigation indicates the presented dual quaternion algorithm (QDA) has an excellent or fast convergence property. If an accurate initial value of scale is provided, e.g., by the solutions no. 2 and 3 of Závoti and Kalmár (Acta Geod Geophys 51: [245][246][247][248][249][250][251][252][253][254][255][256] 2016) in the case that the weights are identical, QDA needs one iteration to obtain the correct result of transformation parameters; in other words, it can be regarded as an analytical algorithm. For other situations, QDA requires two iterations to recover the transformation parameters no matter how big the rotation angles are and how biased the initial value of scale is. Additionally, QDA is capable to deal with point-wise weight transformation which is more rational than those algorithms which simply take identical weights into account or do not consider the weight difference among control points. From the perspective of transformation accuracy, QDA is comparable to the classic Procrustes algorithm (Grafarend and Awange in J Geod 77: [66][67][68][69][70][71][72][73][74][75][76] 2003) and orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015. https://doi.

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The analytical solution of 3D datum transformation with an isotropic weight has been elegantly presented based on Procrustes algorithm (singular value decomposition). But the existence of analytical solution of 3D datum transformation with a nonisotropic weight needs further investigation. Based on the Lagrangian extremum law, the paper derives the analytical formula for translation parameter and scale factor, but because the rotation matrix is unsolved, the analytical solution does not exist. For this reason, the paper presents two kinds of iterative approach of 3D datum transformation with a nonisotropic weight. One is the iterative approach dependent on the objective function value, which uses the Lagrangian minimum function in the variable of rotation matrix as the objective function, and the other is the iterative approach dependent on the derivative of function, which uses the 3D datum transformation model that eliminates the translation parameter. In order to improve the speed and reliability of iterative computation, the form of rotation matrix represented by Rodrigues matrix instead of rotation angles or unit quaternion is adopted for the two iterative approaches. A numerical experiment is demonstrated, and comparison analysis of the two iterative approaches is carried out. The result shows from the view of computing speed and reliability, the iterative approach based on derivatives is preferred.

The 3D similarity coordinate transformation is fundamental and frequently encountered in many areas of work such as geodesy, engineering surveying, LIDAR, terrestrial laser scanning, photogrammetry, machine vision, etc. The algorithms of 3D similarity transformation are divided into two categories. One is a closed-form algorithm that is straightforward and fast. However, it cannot provide the accuracy information for the transformation parameters. The other category of algorithm is iterative, and this can offer the accuracy information for the transformation parameters. However, the latter usually needs a good initial value of the unknown. Considering the accuracy information for transformation parameters is essential or indispensable from the viewpoint of uncertainty, this contribution proposes a weighted total least squares (WTLS) iterative algorithm of the 3D similarity coordinate transformation based on Gibbs vectors. It is fast in terms of fewer iterations, reliable and does not need good initial values of transformation parameters. Two cases including the registration of LIDAR points with big rotation angles and a geodetic datum transformation with small rotation angles are demonstrated to validate the new algorithm.

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